Pretty much forever--so long as you pay your taxes they really don't care where you live.
The United States places no restrictions upon the length of time that its citizens are allowed to remain outside of the US. Citizens can enter or leave the country any time they wish. As for the other (unspecified) country which your daughter is visiting, they may very well have some limit on the length of time for which they allow people to visit, or might require a particular kind of visa. The depends not only on which country she is visiting, but also on the purpose of her visit.
Citizens of the US are not allowed to travel to Cuba under normal circumstances.
To stay within borders
This is a controversial issue. Some people, and certainly the government, seem to believe they should be allowed to stay. Many citizens believe if they entered the country illegally they have already broken the law and should be sent back to their own country; however, there are those who do not agree with that. Some believe they should be granted amnesty and others disagree.
Switzerland is a country that is protected by its mountains. This has allowed for the country to stay neutral in various conflicts.
All foreign national are required to have a visa to enter the US. Even American citizens are required to have a passport to re-enter the country.
Yes, any country can do that. You'd need to contact the government agency that handles immigration in the specific country you're interested in to find out if they do do that, and what you need to do to legally start the process.(The United States, for example, does not. Spouses of citizens become "permanent residents," they are allowed to stay in the country, but they are not citizens and cannot, for example, vote. They must go through the naturalization process if they want to become citizens.)
No more than six months.
It is roughly three weeks by then they know which country is out.
No - but you may need separate visas, depending on the country and your length of stay.
No, US citizens do not normally need a Schengen visa. They are normally allowed to stay for 90 days as a tourist without a visa.
The Union needed to stay together during the U.S. Civil War because it made the country strong. If it was allowed to divide, the country would be much smaller in both size and population.